Properties

Label 1008.2.s.f.865.1
Level $1008$
Weight $2$
Character 1008.865
Analytic conductor $8.049$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,2,Mod(289,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.s (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 865.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1008.865
Dual form 1008.2.s.f.289.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{5} +(-0.500000 - 2.59808i) q^{7} +O(q^{10})\) \(q+(-0.500000 - 0.866025i) q^{5} +(-0.500000 - 2.59808i) q^{7} +(-1.50000 + 2.59808i) q^{11} +4.00000 q^{13} +(-2.00000 - 3.46410i) q^{19} +(-4.00000 - 6.92820i) q^{23} +(2.00000 - 3.46410i) q^{25} +3.00000 q^{29} +(-2.50000 + 4.33013i) q^{31} +(-2.00000 + 1.73205i) q^{35} +(-4.00000 - 6.92820i) q^{37} -8.00000 q^{41} -6.00000 q^{43} +(-5.00000 - 8.66025i) q^{47} +(-6.50000 + 2.59808i) q^{49} +(4.50000 - 7.79423i) q^{53} +3.00000 q^{55} +(2.50000 - 4.33013i) q^{59} +(5.00000 + 8.66025i) q^{61} +(-2.00000 - 3.46410i) q^{65} +(3.00000 - 5.19615i) q^{67} +10.0000 q^{71} +(-1.00000 + 1.73205i) q^{73} +(7.50000 + 2.59808i) q^{77} +(5.50000 + 9.52628i) q^{79} +7.00000 q^{83} +(-9.00000 - 15.5885i) q^{89} +(-2.00000 - 10.3923i) q^{91} +(-2.00000 + 3.46410i) q^{95} -17.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{5} - q^{7} - 3 q^{11} + 8 q^{13} - 4 q^{19} - 8 q^{23} + 4 q^{25} + 6 q^{29} - 5 q^{31} - 4 q^{35} - 8 q^{37} - 16 q^{41} - 12 q^{43} - 10 q^{47} - 13 q^{49} + 9 q^{53} + 6 q^{55} + 5 q^{59} + 10 q^{61} - 4 q^{65} + 6 q^{67} + 20 q^{71} - 2 q^{73} + 15 q^{77} + 11 q^{79} + 14 q^{83} - 18 q^{89} - 4 q^{91} - 4 q^{95} - 34 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −0.500000 0.866025i −0.223607 0.387298i 0.732294 0.680989i \(-0.238450\pi\)
−0.955901 + 0.293691i \(0.905116\pi\)
\(6\) 0 0
\(7\) −0.500000 2.59808i −0.188982 0.981981i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.50000 + 2.59808i −0.452267 + 0.783349i −0.998526 0.0542666i \(-0.982718\pi\)
0.546259 + 0.837616i \(0.316051\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) −2.00000 3.46410i −0.458831 0.794719i 0.540068 0.841621i \(-0.318398\pi\)
−0.998899 + 0.0469020i \(0.985065\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 6.92820i −0.834058 1.44463i −0.894795 0.446476i \(-0.852679\pi\)
0.0607377 0.998154i \(-0.480655\pi\)
\(24\) 0 0
\(25\) 2.00000 3.46410i 0.400000 0.692820i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) −2.50000 + 4.33013i −0.449013 + 0.777714i −0.998322 0.0579057i \(-0.981558\pi\)
0.549309 + 0.835619i \(0.314891\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.00000 + 1.73205i −0.338062 + 0.292770i
\(36\) 0 0
\(37\) −4.00000 6.92820i −0.657596 1.13899i −0.981236 0.192809i \(-0.938240\pi\)
0.323640 0.946180i \(-0.395093\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.00000 8.66025i −0.729325 1.26323i −0.957169 0.289530i \(-0.906501\pi\)
0.227844 0.973698i \(-0.426832\pi\)
\(48\) 0 0
\(49\) −6.50000 + 2.59808i −0.928571 + 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.50000 7.79423i 0.618123 1.07062i −0.371706 0.928351i \(-0.621227\pi\)
0.989828 0.142269i \(-0.0454398\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.50000 4.33013i 0.325472 0.563735i −0.656136 0.754643i \(-0.727810\pi\)
0.981608 + 0.190909i \(0.0611434\pi\)
\(60\) 0 0
\(61\) 5.00000 + 8.66025i 0.640184 + 1.10883i 0.985391 + 0.170305i \(0.0544754\pi\)
−0.345207 + 0.938527i \(0.612191\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.00000 3.46410i −0.248069 0.429669i
\(66\) 0 0
\(67\) 3.00000 5.19615i 0.366508 0.634811i −0.622509 0.782613i \(-0.713886\pi\)
0.989017 + 0.147802i \(0.0472198\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.0000 1.18678 0.593391 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) 0 0
\(73\) −1.00000 + 1.73205i −0.117041 + 0.202721i −0.918594 0.395203i \(-0.870674\pi\)
0.801553 + 0.597924i \(0.204008\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 7.50000 + 2.59808i 0.854704 + 0.296078i
\(78\) 0 0
\(79\) 5.50000 + 9.52628i 0.618798 + 1.07179i 0.989705 + 0.143120i \(0.0457135\pi\)
−0.370907 + 0.928670i \(0.620953\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.00000 0.768350 0.384175 0.923260i \(-0.374486\pi\)
0.384175 + 0.923260i \(0.374486\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −9.00000 15.5885i −0.953998 1.65237i −0.736644 0.676280i \(-0.763591\pi\)
−0.217354 0.976093i \(-0.569742\pi\)
\(90\) 0 0
\(91\) −2.00000 10.3923i −0.209657 1.08941i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.00000 + 3.46410i −0.205196 + 0.355409i
\(96\) 0 0
\(97\) −17.0000 −1.72609 −0.863044 0.505128i \(-0.831445\pi\)
−0.863044 + 0.505128i \(0.831445\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.00000 + 1.73205i −0.0995037 + 0.172345i −0.911479 0.411346i \(-0.865059\pi\)
0.811976 + 0.583691i \(0.198392\pi\)
\(102\) 0 0
\(103\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.50000 + 9.52628i 0.531705 + 0.920940i 0.999315 + 0.0370053i \(0.0117818\pi\)
−0.467610 + 0.883935i \(0.654885\pi\)
\(108\) 0 0
\(109\) 5.00000 8.66025i 0.478913 0.829502i −0.520794 0.853682i \(-0.674364\pi\)
0.999708 + 0.0241802i \(0.00769755\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 8.00000 0.752577 0.376288 0.926503i \(-0.377200\pi\)
0.376288 + 0.926503i \(0.377200\pi\)
\(114\) 0 0
\(115\) −4.00000 + 6.92820i −0.373002 + 0.646058i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 + 1.73205i 0.0909091 + 0.157459i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) 7.00000 0.621150 0.310575 0.950549i \(-0.399478\pi\)
0.310575 + 0.950549i \(0.399478\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −7.50000 12.9904i −0.655278 1.13497i −0.981824 0.189794i \(-0.939218\pi\)
0.326546 0.945181i \(-0.394115\pi\)
\(132\) 0 0
\(133\) −8.00000 + 6.92820i −0.693688 + 0.600751i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −7.00000 + 12.1244i −0.598050 + 1.03585i 0.395058 + 0.918656i \(0.370724\pi\)
−0.993109 + 0.117198i \(0.962609\pi\)
\(138\) 0 0
\(139\) 22.0000 1.86602 0.933008 0.359856i \(-0.117174\pi\)
0.933008 + 0.359856i \(0.117174\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −6.00000 + 10.3923i −0.501745 + 0.869048i
\(144\) 0 0
\(145\) −1.50000 2.59808i −0.124568 0.215758i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 9.00000 + 15.5885i 0.737309 + 1.27706i 0.953703 + 0.300750i \(0.0972370\pi\)
−0.216394 + 0.976306i \(0.569430\pi\)
\(150\) 0 0
\(151\) −2.50000 + 4.33013i −0.203447 + 0.352381i −0.949637 0.313353i \(-0.898548\pi\)
0.746190 + 0.665733i \(0.231881\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 5.00000 0.401610
\(156\) 0 0
\(157\) 2.00000 3.46410i 0.159617 0.276465i −0.775113 0.631822i \(-0.782307\pi\)
0.934731 + 0.355357i \(0.115641\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −16.0000 + 13.8564i −1.26098 + 1.09204i
\(162\) 0 0
\(163\) 4.00000 + 6.92820i 0.313304 + 0.542659i 0.979076 0.203497i \(-0.0652307\pi\)
−0.665771 + 0.746156i \(0.731897\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.00000 −0.154765 −0.0773823 0.997001i \(-0.524656\pi\)
−0.0773823 + 0.997001i \(0.524656\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.00000 + 1.73205i 0.0760286 + 0.131685i 0.901533 0.432710i \(-0.142443\pi\)
−0.825505 + 0.564396i \(0.809109\pi\)
\(174\) 0 0
\(175\) −10.0000 3.46410i −0.755929 0.261861i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −6.00000 + 10.3923i −0.448461 + 0.776757i −0.998286 0.0585225i \(-0.981361\pi\)
0.549825 + 0.835280i \(0.314694\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −4.00000 + 6.92820i −0.294086 + 0.509372i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 8.00000 + 13.8564i 0.578860 + 1.00261i 0.995610 + 0.0935936i \(0.0298354\pi\)
−0.416751 + 0.909021i \(0.636831\pi\)
\(192\) 0 0
\(193\) 13.5000 23.3827i 0.971751 1.68312i 0.281487 0.959565i \(-0.409172\pi\)
0.690264 0.723558i \(-0.257494\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 26.0000 1.85242 0.926212 0.377004i \(-0.123046\pi\)
0.926212 + 0.377004i \(0.123046\pi\)
\(198\) 0 0
\(199\) 6.00000 10.3923i 0.425329 0.736691i −0.571122 0.820865i \(-0.693492\pi\)
0.996451 + 0.0841740i \(0.0268252\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1.50000 7.79423i −0.105279 0.547048i
\(204\) 0 0
\(205\) 4.00000 + 6.92820i 0.279372 + 0.483887i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) −6.00000 −0.413057 −0.206529 0.978441i \(-0.566217\pi\)
−0.206529 + 0.978441i \(0.566217\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3.00000 + 5.19615i 0.204598 + 0.354375i
\(216\) 0 0
\(217\) 12.5000 + 4.33013i 0.848555 + 0.293948i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.00000 −0.0669650 −0.0334825 0.999439i \(-0.510660\pi\)
−0.0334825 + 0.999439i \(0.510660\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −10.5000 + 18.1865i −0.696909 + 1.20708i 0.272623 + 0.962121i \(0.412109\pi\)
−0.969533 + 0.244962i \(0.921225\pi\)
\(228\) 0 0
\(229\) 6.00000 + 10.3923i 0.396491 + 0.686743i 0.993290 0.115648i \(-0.0368944\pi\)
−0.596799 + 0.802391i \(0.703561\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −12.0000 20.7846i −0.786146 1.36165i −0.928312 0.371802i \(-0.878740\pi\)
0.142166 0.989843i \(-0.454593\pi\)
\(234\) 0 0
\(235\) −5.00000 + 8.66025i −0.326164 + 0.564933i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −11.5000 + 19.9186i −0.740780 + 1.28307i 0.211360 + 0.977408i \(0.432211\pi\)
−0.952141 + 0.305661i \(0.901123\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 5.50000 + 4.33013i 0.351382 + 0.276642i
\(246\) 0 0
\(247\) −8.00000 13.8564i −0.509028 0.881662i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 11.0000 0.694314 0.347157 0.937807i \(-0.387147\pi\)
0.347157 + 0.937807i \(0.387147\pi\)
\(252\) 0 0
\(253\) 24.0000 1.50887
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7.00000 + 12.1244i 0.436648 + 0.756297i 0.997429 0.0716680i \(-0.0228322\pi\)
−0.560781 + 0.827964i \(0.689499\pi\)
\(258\) 0 0
\(259\) −16.0000 + 13.8564i −0.994192 + 0.860995i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 7.00000 12.1244i 0.431638 0.747620i −0.565376 0.824833i \(-0.691269\pi\)
0.997015 + 0.0772134i \(0.0246023\pi\)
\(264\) 0 0
\(265\) −9.00000 −0.552866
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0.500000 0.866025i 0.0304855 0.0528025i −0.850380 0.526169i \(-0.823628\pi\)
0.880866 + 0.473366i \(0.156961\pi\)
\(270\) 0 0
\(271\) −8.50000 14.7224i −0.516338 0.894324i −0.999820 0.0189696i \(-0.993961\pi\)
0.483482 0.875354i \(-0.339372\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 6.00000 + 10.3923i 0.361814 + 0.626680i
\(276\) 0 0
\(277\) −4.00000 + 6.92820i −0.240337 + 0.416275i −0.960810 0.277207i \(-0.910591\pi\)
0.720473 + 0.693482i \(0.243925\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) −9.00000 + 15.5885i −0.534994 + 0.926638i 0.464169 + 0.885747i \(0.346353\pi\)
−0.999164 + 0.0408910i \(0.986980\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.00000 + 20.7846i 0.236113 + 1.22688i
\(288\) 0 0
\(289\) 8.50000 + 14.7224i 0.500000 + 0.866025i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 19.0000 1.10999 0.554996 0.831853i \(-0.312720\pi\)
0.554996 + 0.831853i \(0.312720\pi\)
\(294\) 0 0
\(295\) −5.00000 −0.291111
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −16.0000 27.7128i −0.925304 1.60267i
\(300\) 0 0
\(301\) 3.00000 + 15.5885i 0.172917 + 0.898504i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 5.00000 8.66025i 0.286299 0.495885i
\(306\) 0 0
\(307\) 16.0000 0.913168 0.456584 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(312\) 0 0
\(313\) −4.50000 7.79423i −0.254355 0.440556i 0.710365 0.703833i \(-0.248530\pi\)
−0.964720 + 0.263278i \(0.915197\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.50000 + 11.2583i 0.365076 + 0.632331i 0.988788 0.149323i \(-0.0477095\pi\)
−0.623712 + 0.781654i \(0.714376\pi\)
\(318\) 0 0
\(319\) −4.50000 + 7.79423i −0.251952 + 0.436393i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 8.00000 13.8564i 0.443760 0.768615i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −20.0000 + 17.3205i −1.10264 + 0.954911i
\(330\) 0 0
\(331\) 2.00000 + 3.46410i 0.109930 + 0.190404i 0.915742 0.401768i \(-0.131604\pi\)
−0.805812 + 0.592172i \(0.798271\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −6.00000 −0.327815
\(336\) 0 0
\(337\) −15.0000 −0.817102 −0.408551 0.912735i \(-0.633966\pi\)
−0.408551 + 0.912735i \(0.633966\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −7.50000 12.9904i −0.406148 0.703469i
\(342\) 0 0
\(343\) 10.0000 + 15.5885i 0.539949 + 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 10.0000 17.3205i 0.536828 0.929814i −0.462244 0.886753i \(-0.652956\pi\)
0.999072 0.0430610i \(-0.0137110\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 4.00000 6.92820i 0.212899 0.368751i −0.739722 0.672913i \(-0.765043\pi\)
0.952620 + 0.304162i \(0.0983763\pi\)
\(354\) 0 0
\(355\) −5.00000 8.66025i −0.265372 0.459639i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −3.00000 5.19615i −0.158334 0.274242i 0.775934 0.630814i \(-0.217279\pi\)
−0.934268 + 0.356572i \(0.883946\pi\)
\(360\) 0 0
\(361\) 1.50000 2.59808i 0.0789474 0.136741i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2.00000 0.104685
\(366\) 0 0
\(367\) −3.50000 + 6.06218i −0.182699 + 0.316443i −0.942799 0.333363i \(-0.891817\pi\)
0.760100 + 0.649806i \(0.225150\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −22.5000 7.79423i −1.16814 0.404656i
\(372\) 0 0
\(373\) −8.00000 13.8564i −0.414224 0.717458i 0.581122 0.813816i \(-0.302614\pi\)
−0.995347 + 0.0963587i \(0.969280\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.00000 1.73205i −0.0510976 0.0885037i 0.839345 0.543599i \(-0.182939\pi\)
−0.890443 + 0.455095i \(0.849605\pi\)
\(384\) 0 0
\(385\) −1.50000 7.79423i −0.0764471 0.397231i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 13.0000 22.5167i 0.659126 1.14164i −0.321716 0.946836i \(-0.604260\pi\)
0.980842 0.194804i \(-0.0624070\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 5.50000 9.52628i 0.276735 0.479319i
\(396\) 0 0
\(397\) −14.0000 24.2487i −0.702640 1.21701i −0.967537 0.252731i \(-0.918671\pi\)
0.264897 0.964277i \(-0.414662\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −4.00000 6.92820i −0.199750 0.345978i 0.748697 0.662912i \(-0.230680\pi\)
−0.948447 + 0.316934i \(0.897346\pi\)
\(402\) 0 0
\(403\) −10.0000 + 17.3205i −0.498135 + 0.862796i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 24.0000 1.18964
\(408\) 0 0
\(409\) −3.50000 + 6.06218i −0.173064 + 0.299755i −0.939490 0.342578i \(-0.888700\pi\)
0.766426 + 0.642333i \(0.222033\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −12.5000 4.33013i −0.615085 0.213072i
\(414\) 0 0
\(415\) −3.50000 6.06218i −0.171808 0.297581i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 20.0000 17.3205i 0.967868 0.838198i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 6.00000 10.3923i 0.289010 0.500580i −0.684564 0.728953i \(-0.740007\pi\)
0.973574 + 0.228373i \(0.0733406\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −16.0000 + 27.7128i −0.765384 + 1.32568i
\(438\) 0 0
\(439\) −4.50000 7.79423i −0.214773 0.371998i 0.738429 0.674331i \(-0.235568\pi\)
−0.953202 + 0.302333i \(0.902235\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0.500000 + 0.866025i 0.0237557 + 0.0411461i 0.877659 0.479286i \(-0.159104\pi\)
−0.853903 + 0.520432i \(0.825771\pi\)
\(444\) 0 0
\(445\) −9.00000 + 15.5885i −0.426641 + 0.738964i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −4.00000 −0.188772 −0.0943858 0.995536i \(-0.530089\pi\)
−0.0943858 + 0.995536i \(0.530089\pi\)
\(450\) 0 0
\(451\) 12.0000 20.7846i 0.565058 0.978709i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −8.00000 + 6.92820i −0.375046 + 0.324799i
\(456\) 0 0
\(457\) 8.50000 + 14.7224i 0.397613 + 0.688686i 0.993431 0.114433i \(-0.0365053\pi\)
−0.595818 + 0.803120i \(0.703172\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 14.0000 + 24.2487i 0.647843 + 1.12210i 0.983637 + 0.180161i \(0.0576619\pi\)
−0.335794 + 0.941935i \(0.609005\pi\)
\(468\) 0 0
\(469\) −15.0000 5.19615i −0.692636 0.239936i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 9.00000 15.5885i 0.413820 0.716758i
\(474\) 0 0
\(475\) −16.0000 −0.734130
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −3.00000 + 5.19615i −0.137073 + 0.237418i −0.926388 0.376571i \(-0.877103\pi\)
0.789314 + 0.613990i \(0.210436\pi\)
\(480\) 0 0
\(481\) −16.0000 27.7128i −0.729537 1.26360i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 8.50000 + 14.7224i 0.385965 + 0.668511i
\(486\) 0 0
\(487\) 18.5000 32.0429i 0.838315 1.45200i −0.0529875 0.998595i \(-0.516874\pi\)
0.891303 0.453409i \(-0.149792\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −9.00000 −0.406164 −0.203082 0.979162i \(-0.565096\pi\)
−0.203082 + 0.979162i \(0.565096\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −5.00000 25.9808i −0.224281 1.16540i
\(498\) 0 0
\(499\) 5.00000 + 8.66025i 0.223831 + 0.387686i 0.955968 0.293471i \(-0.0948104\pi\)
−0.732137 + 0.681157i \(0.761477\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) 2.00000 0.0889988
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −19.5000 33.7750i −0.864322 1.49705i −0.867719 0.497056i \(-0.834414\pi\)
0.00339621 0.999994i \(-0.498919\pi\)
\(510\) 0 0
\(511\) 5.00000 + 1.73205i 0.221187 + 0.0766214i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 30.0000 1.31940
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −9.00000 + 15.5885i −0.394297 + 0.682943i −0.993011 0.118020i \(-0.962345\pi\)
0.598714 + 0.800963i \(0.295679\pi\)
\(522\) 0 0
\(523\) −6.00000 10.3923i −0.262362 0.454424i 0.704507 0.709697i \(-0.251168\pi\)
−0.966869 + 0.255273i \(0.917835\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −20.5000 + 35.5070i −0.891304 + 1.54378i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −32.0000 −1.38607
\(534\) 0 0
\(535\) 5.50000 9.52628i 0.237786 0.411857i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 3.00000 20.7846i 0.129219 0.895257i
\(540\) 0 0
\(541\) −17.0000 29.4449i −0.730887 1.26593i −0.956504 0.291718i \(-0.905773\pi\)
0.225617 0.974216i \(-0.427560\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −6.00000 10.3923i −0.255609 0.442727i
\(552\) 0 0
\(553\) 22.0000 19.0526i 0.935535 0.810197i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 7.50000 12.9904i 0.317785 0.550420i −0.662240 0.749291i \(-0.730394\pi\)
0.980026 + 0.198871i \(0.0637276\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −19.5000 + 33.7750i −0.821827 + 1.42345i 0.0824933 + 0.996592i \(0.473712\pi\)
−0.904320 + 0.426855i \(0.859622\pi\)
\(564\) 0 0
\(565\) −4.00000 6.92820i −0.168281 0.291472i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 6.00000 + 10.3923i 0.251533 + 0.435668i 0.963948 0.266090i \(-0.0857319\pi\)
−0.712415 + 0.701758i \(0.752399\pi\)
\(570\) 0 0
\(571\) 11.0000 19.0526i 0.460336 0.797325i −0.538642 0.842535i \(-0.681062\pi\)
0.998978 + 0.0452101i \(0.0143957\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −32.0000 −1.33449
\(576\) 0 0
\(577\) 16.5000 28.5788i 0.686904 1.18975i −0.285930 0.958250i \(-0.592303\pi\)
0.972834 0.231502i \(-0.0743641\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −3.50000 18.1865i −0.145204 0.754505i
\(582\) 0 0
\(583\) 13.5000 + 23.3827i 0.559113 + 0.968412i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 45.0000 1.85735 0.928674 0.370896i \(-0.120949\pi\)
0.928674 + 0.370896i \(0.120949\pi\)
\(588\) 0 0
\(589\) 20.0000 0.824086
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 21.0000 36.3731i 0.858037 1.48616i −0.0157622 0.999876i \(-0.505017\pi\)
0.873799 0.486287i \(-0.161649\pi\)
\(600\) 0 0
\(601\) −13.0000 −0.530281 −0.265141 0.964210i \(-0.585418\pi\)
−0.265141 + 0.964210i \(0.585418\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.00000 1.73205i 0.0406558 0.0704179i
\(606\) 0 0
\(607\) 6.50000 + 11.2583i 0.263827 + 0.456962i 0.967256 0.253804i \(-0.0816819\pi\)
−0.703429 + 0.710766i \(0.748349\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −20.0000 34.6410i −0.809113 1.40143i
\(612\) 0 0
\(613\) −4.00000 + 6.92820i −0.161558 + 0.279827i −0.935428 0.353518i \(-0.884985\pi\)
0.773869 + 0.633345i \(0.218319\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) 3.00000 5.19615i 0.120580 0.208851i −0.799416 0.600777i \(-0.794858\pi\)
0.919997 + 0.391926i \(0.128191\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −36.0000 + 31.1769i −1.44231 + 1.24908i
\(624\) 0 0
\(625\) −5.50000 9.52628i −0.220000 0.381051i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 11.0000 0.437903 0.218952 0.975736i \(-0.429736\pi\)
0.218952 + 0.975736i \(0.429736\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −3.50000 6.06218i −0.138893 0.240570i
\(636\) 0 0
\(637\) −26.0000 + 10.3923i −1.03016 + 0.411758i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −11.0000 + 19.0526i −0.434474 + 0.752531i −0.997253 0.0740768i \(-0.976399\pi\)
0.562779 + 0.826608i \(0.309732\pi\)
\(642\) 0 0
\(643\) −46.0000 −1.81406 −0.907031 0.421063i \(-0.861657\pi\)
−0.907031 + 0.421063i \(0.861657\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 25.0000 43.3013i 0.982851 1.70235i 0.331731 0.943374i \(-0.392367\pi\)
0.651120 0.758975i \(-0.274299\pi\)
\(648\) 0 0
\(649\) 7.50000 + 12.9904i 0.294401 + 0.509917i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 7.50000 + 12.9904i 0.293498 + 0.508353i 0.974634 0.223803i \(-0.0718474\pi\)
−0.681137 + 0.732156i \(0.738514\pi\)
\(654\) 0 0
\(655\) −7.50000 + 12.9904i −0.293049 + 0.507576i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 24.0000 0.934907 0.467454 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) 0 0
\(661\) −19.0000 + 32.9090i −0.739014 + 1.28001i 0.213925 + 0.976850i \(0.431375\pi\)
−0.952940 + 0.303160i \(0.901958\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 10.0000 + 3.46410i 0.387783 + 0.134332i
\(666\) 0 0
\(667\) −12.0000 20.7846i −0.464642 0.804783i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −30.0000 −1.15814
\(672\) 0 0
\(673\) 29.0000 1.11787 0.558934 0.829212i \(-0.311211\pi\)
0.558934 + 0.829212i \(0.311211\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 9.50000 + 16.4545i 0.365115 + 0.632397i 0.988795 0.149283i \(-0.0476964\pi\)
−0.623680 + 0.781680i \(0.714363\pi\)
\(678\) 0 0
\(679\) 8.50000 + 44.1673i 0.326200 + 1.69499i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −8.50000 + 14.7224i −0.325243 + 0.563338i −0.981562 0.191146i \(-0.938780\pi\)
0.656318 + 0.754484i \(0.272113\pi\)
\(684\) 0 0
\(685\) 14.0000 0.534913
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 18.0000 31.1769i 0.685745 1.18775i
\(690\) 0 0
\(691\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −11.0000 19.0526i −0.417254 0.722705i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −5.00000 −0.188847 −0.0944237 0.995532i \(-0.530101\pi\)
−0.0944237 + 0.995532i \(0.530101\pi\)
\(702\) 0 0
\(703\) −16.0000 + 27.7128i −0.603451 + 1.04521i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 5.00000 + 1.73205i 0.188044 + 0.0651405i
\(708\) 0 0
\(709\) −3.00000 5.19615i −0.112667 0.195146i 0.804178 0.594389i \(-0.202606\pi\)
−0.916845 + 0.399244i \(0.869273\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 40.0000 1.49801
\(714\) 0 0
\(715\) 12.0000 0.448775
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 17.0000 + 29.4449i 0.633993 + 1.09811i 0.986728 + 0.162385i \(0.0519185\pi\)
−0.352735 + 0.935723i \(0.614748\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 6.00000 10.3923i 0.222834 0.385961i
\(726\) 0 0
\(727\) −47.0000 −1.74313 −0.871567 0.490277i \(-0.836896\pi\)
−0.871567 + 0.490277i \(0.836896\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −7.00000 12.1244i −0.258551 0.447823i 0.707303 0.706910i \(-0.249912\pi\)
−0.965854 + 0.259087i \(0.916578\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 9.00000 + 15.5885i 0.331519 + 0.574208i
\(738\) 0 0
\(739\) 5.00000 8.66025i 0.183928 0.318573i −0.759287 0.650756i \(-0.774452\pi\)
0.943215 + 0.332184i \(0.107785\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 34.0000 1.24734 0.623670 0.781688i \(-0.285641\pi\)
0.623670 + 0.781688i \(0.285641\pi\)
\(744\) 0 0
\(745\) 9.00000 15.5885i 0.329734 0.571117i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 22.0000 19.0526i 0.803863 0.696165i
\(750\) 0 0
\(751\) −13.5000 23.3827i −0.492622 0.853246i 0.507342 0.861745i \(-0.330628\pi\)
−0.999964 + 0.00849853i \(0.997295\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 5.00000 0.181969
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −6.00000 10.3923i −0.217500 0.376721i 0.736543 0.676391i \(-0.236457\pi\)
−0.954043 + 0.299670i \(0.903123\pi\)
\(762\) 0 0
\(763\) −25.0000 8.66025i −0.905061 0.313522i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 10.0000 17.3205i 0.361079 0.625407i
\(768\) 0 0
\(769\) 37.0000 1.33425 0.667127 0.744944i \(-0.267524\pi\)
0.667127 + 0.744944i \(0.267524\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −13.0000 + 22.5167i −0.467578 + 0.809868i −0.999314 0.0370420i \(-0.988206\pi\)
0.531736 + 0.846910i \(0.321540\pi\)
\(774\) 0 0
\(775\) 10.0000 + 17.3205i 0.359211 + 0.622171i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 16.0000 + 27.7128i 0.573259 + 0.992915i
\(780\) 0 0
\(781\) −15.0000 + 25.9808i −0.536742 + 0.929665i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −4.00000 −0.142766
\(786\) 0 0
\(787\) −11.0000 + 19.0526i −0.392108 + 0.679150i −0.992727 0.120384i \(-0.961587\pi\)
0.600620 + 0.799535i \(0.294921\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −4.00000 20.7846i −0.142224 0.739016i
\(792\) 0 0
\(793\) 20.0000 + 34.6410i 0.710221 + 1.23014i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −3.00000 −0.106265 −0.0531327 0.998587i \(-0.516921\pi\)
−0.0531327 + 0.998587i \(0.516921\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −3.00000 5.19615i −0.105868 0.183368i
\(804\) 0 0
\(805\) 20.0000 + 6.92820i 0.704907 + 0.244187i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −8.00000 + 13.8564i −0.281265 + 0.487165i −0.971697 0.236232i \(-0.924087\pi\)
0.690432 + 0.723398i \(0.257421\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 4.00000 6.92820i 0.140114 0.242684i
\(816\) 0 0
\(817\) 12.0000 + 20.7846i 0.419827 + 0.727161i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −19.5000 33.7750i −0.680555 1.17876i −0.974812 0.223029i \(-0.928406\pi\)
0.294257 0.955726i \(-0.404928\pi\)
\(822\) 0 0
\(823\) 16.0000 27.7128i 0.557725 0.966008i −0.439961 0.898017i \(-0.645008\pi\)
0.997686 0.0679910i \(-0.0216589\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −41.0000 −1.42571 −0.712855 0.701312i \(-0.752598\pi\)
−0.712855 + 0.701312i \(0.752598\pi\)
\(828\) 0 0
\(829\) 2.00000 3.46410i 0.0694629 0.120313i −0.829202 0.558949i \(-0.811205\pi\)
0.898665 + 0.438636i \(0.144538\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 1.00000 + 1.73205i 0.0346064 + 0.0599401i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1.50000 2.59808i −0.0516016 0.0893765i
\(846\) 0 0
\(847\) 4.00000 3.46410i 0.137442 0.119028i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −32.0000 + 55.4256i −1.09695 + 1.89997i
\(852\) 0 0
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 7.00000 12.1244i 0.239115 0.414160i −0.721345 0.692576i \(-0.756476\pi\)
0.960461 + 0.278416i \(0.0898092\pi\)
\(858\) 0 0
\(859\) −7.00000 12.1244i −0.238837 0.413678i 0.721544 0.692369i \(-0.243433\pi\)
−0.960381 + 0.278691i \(0.910099\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −11.0000 19.0526i −0.374444 0.648557i 0.615799 0.787903i \(-0.288833\pi\)
−0.990244 + 0.139346i \(0.955500\pi\)
\(864\) 0 0
\(865\) 1.00000 1.73205i 0.0340010 0.0588915i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −33.0000 −1.11945
\(870\) 0 0
\(871\) 12.0000 20.7846i 0.406604 0.704260i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 4.50000 + 23.3827i 0.152128 + 0.790479i
\(876\) 0 0
\(877\) 16.0000 + 27.7128i 0.540282 + 0.935795i 0.998888 + 0.0471555i \(0.0150156\pi\)
−0.458606 + 0.888640i \(0.651651\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 54.0000 1.81931 0.909653 0.415369i \(-0.136347\pi\)
0.909653 + 0.415369i \(0.136347\pi\)
\(882\) 0 0
\(883\) −40.0000 −1.34611 −0.673054 0.739594i \(-0.735018\pi\)
−0.673054 + 0.739594i \(0.735018\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 24.0000 + 41.5692i 0.805841 + 1.39576i 0.915722 + 0.401813i \(0.131620\pi\)
−0.109881 + 0.993945i \(0.535047\pi\)
\(888\) 0 0
\(889\) −3.50000 18.1865i −0.117386 0.609957i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −20.0000 + 34.6410i −0.669274 + 1.15922i
\(894\) 0 0
\(895\) 12.0000 0.401116
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −7.50000 + 12.9904i −0.250139 + 0.433253i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −8.00000 + 13.8564i −0.265636 + 0.460094i −0.967730 0.251990i \(-0.918915\pi\)
0.702094 + 0.712084i \(0.252248\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 6.00000 0.198789 0.0993944 0.995048i \(-0.468309\pi\)
0.0993944 + 0.995048i \(0.468309\pi\)
\(912\) 0 0
\(913\) −10.5000 + 18.1865i −0.347499 + 0.601886i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −30.0000 + 25.9808i −0.990687 + 0.857960i
\(918\) 0 0
\(919\) −8.00000 13.8564i −0.263896 0.457081i 0.703378 0.710816i \(-0.251674\pi\)
−0.967274 + 0.253735i \(0.918341\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 40.0000 1.31662
\(924\) 0 0
\(925\) −32.0000 −1.05215
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 21.0000 + 36.3731i 0.688988 + 1.19336i 0.972166 + 0.234294i \(0.0752779\pi\)
−0.283178 + 0.959067i \(0.591389\pi\)
\(930\) 0 0
\(931\) 22.0000 + 17.3205i 0.721021 + 0.567657i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 19.0000 0.620703 0.310351 0.950622i \(-0.399553\pi\)
0.310351 + 0.950622i \(0.399553\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −6.50000 + 11.2583i −0.211894 + 0.367011i −0.952307 0.305141i \(-0.901296\pi\)
0.740413 + 0.672152i \(0.234630\pi\)
\(942\) 0 0
\(943\) 32.0000 + 55.4256i 1.04206 + 1.80491i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 24.0000 + 41.5692i 0.779895 + 1.35082i 0.932002 + 0.362454i \(0.118061\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(948\) 0 0
\(949\) −4.00000 + 6.92820i −0.129845 + 0.224899i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 38.0000 1.23094 0.615470 0.788160i \(-0.288966\pi\)
0.615470 + 0.788160i \(0.288966\pi\)
\(954\) 0 0
\(955\) 8.00000 13.8564i 0.258874 0.448383i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 35.0000 + 12.1244i 1.13021 + 0.391516i
\(960\) 0 0
\(961\) 3.00000 + 5.19615i 0.0967742 + 0.167618i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −27.0000 −0.869161
\(966\) 0 0
\(967\) −3.00000 −0.0964735 −0.0482367 0.998836i \(-0.515360\pi\)
−0.0482367 + 0.998836i \(0.515360\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −12.5000 21.6506i −0.401144 0.694802i 0.592720 0.805408i \(-0.298054\pi\)
−0.993864 + 0.110607i \(0.964721\pi\)
\(972\) 0 0
\(973\) −11.0000 57.1577i −0.352644 1.83239i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −11.0000 + 19.0526i −0.351921 + 0.609545i −0.986586 0.163242i \(-0.947805\pi\)
0.634665 + 0.772787i \(0.281138\pi\)
\(978\) 0 0
\(979\) 54.0000 1.72585
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −22.0000 + 38.1051i −0.701691 + 1.21536i 0.266181 + 0.963923i \(0.414238\pi\)
−0.967872 + 0.251442i \(0.919095\pi\)
\(984\) 0 0
\(985\) −13.0000 22.5167i −0.414214 0.717440i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 24.0000 + 41.5692i 0.763156 + 1.32182i
\(990\) 0 0
\(991\) 7.50000 12.9904i 0.238245 0.412653i −0.721966 0.691929i \(-0.756761\pi\)
0.960211 + 0.279276i \(0.0900944\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −12.0000 −0.380426
\(996\) 0 0
\(997\) 13.0000 22.5167i 0.411714 0.713110i −0.583363 0.812211i \(-0.698264\pi\)
0.995077 + 0.0991016i \(0.0315969\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1008.2.s.f.865.1 2
3.2 odd 2 336.2.q.e.193.1 2
4.3 odd 2 504.2.s.d.361.1 2
7.2 even 3 inner 1008.2.s.f.289.1 2
7.3 odd 6 7056.2.a.t.1.1 1
7.4 even 3 7056.2.a.bk.1.1 1
12.11 even 2 168.2.q.a.25.1 2
21.2 odd 6 336.2.q.e.289.1 2
21.5 even 6 2352.2.q.f.961.1 2
21.11 odd 6 2352.2.a.g.1.1 1
21.17 even 6 2352.2.a.u.1.1 1
21.20 even 2 2352.2.q.f.1537.1 2
24.5 odd 2 1344.2.q.d.193.1 2
24.11 even 2 1344.2.q.o.193.1 2
28.3 even 6 3528.2.a.i.1.1 1
28.11 odd 6 3528.2.a.q.1.1 1
28.19 even 6 3528.2.s.p.3313.1 2
28.23 odd 6 504.2.s.d.289.1 2
28.27 even 2 3528.2.s.p.361.1 2
84.11 even 6 1176.2.a.g.1.1 1
84.23 even 6 168.2.q.a.121.1 yes 2
84.47 odd 6 1176.2.q.g.961.1 2
84.59 odd 6 1176.2.a.c.1.1 1
84.83 odd 2 1176.2.q.g.361.1 2
168.11 even 6 9408.2.a.ba.1.1 1
168.53 odd 6 9408.2.a.cq.1.1 1
168.59 odd 6 9408.2.a.cf.1.1 1
168.101 even 6 9408.2.a.p.1.1 1
168.107 even 6 1344.2.q.o.961.1 2
168.149 odd 6 1344.2.q.d.961.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.q.a.25.1 2 12.11 even 2
168.2.q.a.121.1 yes 2 84.23 even 6
336.2.q.e.193.1 2 3.2 odd 2
336.2.q.e.289.1 2 21.2 odd 6
504.2.s.d.289.1 2 28.23 odd 6
504.2.s.d.361.1 2 4.3 odd 2
1008.2.s.f.289.1 2 7.2 even 3 inner
1008.2.s.f.865.1 2 1.1 even 1 trivial
1176.2.a.c.1.1 1 84.59 odd 6
1176.2.a.g.1.1 1 84.11 even 6
1176.2.q.g.361.1 2 84.83 odd 2
1176.2.q.g.961.1 2 84.47 odd 6
1344.2.q.d.193.1 2 24.5 odd 2
1344.2.q.d.961.1 2 168.149 odd 6
1344.2.q.o.193.1 2 24.11 even 2
1344.2.q.o.961.1 2 168.107 even 6
2352.2.a.g.1.1 1 21.11 odd 6
2352.2.a.u.1.1 1 21.17 even 6
2352.2.q.f.961.1 2 21.5 even 6
2352.2.q.f.1537.1 2 21.20 even 2
3528.2.a.i.1.1 1 28.3 even 6
3528.2.a.q.1.1 1 28.11 odd 6
3528.2.s.p.361.1 2 28.27 even 2
3528.2.s.p.3313.1 2 28.19 even 6
7056.2.a.t.1.1 1 7.3 odd 6
7056.2.a.bk.1.1 1 7.4 even 3
9408.2.a.p.1.1 1 168.101 even 6
9408.2.a.ba.1.1 1 168.11 even 6
9408.2.a.cf.1.1 1 168.59 odd 6
9408.2.a.cq.1.1 1 168.53 odd 6